

A084386


Number of pairs of rabbits when there are 3 pairs per litter and offspring reach parenthood after 3 gestation periods; a(n) = a(n1) + 3*a(n3), with a(0) = a(1) = a(2) = 1.


9



1, 1, 1, 4, 7, 10, 22, 43, 73, 139, 268, 487, 904, 1708, 3169, 5881, 11005, 20512, 38155, 71170, 132706, 247171, 460681, 858799, 1600312, 2982355, 5558752, 10359688, 19306753, 35983009, 67062073, 124982332, 232931359, 434117578, 809064574
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


COMMENTS

This comment covers an infinite family of growth sequences, where a(n) = a(n1) + k*a(nm). k is number of pairs per litter and m is periods until adulthood. G.f. = 1/(1xk*x^m). For example, A000930 has k=1 and m=3 while A006130 has k=3 and m=2.
The compositions of n in which each natural number is colored by one of p different colors are called pcolored compositions of n. For n>=3, 4*a(n3) equals the number of 4colored compositions of n with all parts >=3, such that no adjacent parts have the same color.  Milan Janjic, Nov 27 2011
a(n+2) equals the number of words of length n on alphabet {0,1,2,3}, having at least two zeros between every two successive nonzero letters.  Milan Janjic, Feb 07 2015
Number of compositions of n into one sort of part 1 and three sorts of part 3 (see the g.f.).  Joerg Arndt, Feb 07 2015


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Merrill Jensen, Generating Functions
Index entries for linear recurrences with constant coefficients, signature (1,0,3).


FORMULA

a(n) = a(n1) + 3*a(n3). a(n) = A052900(n+3)/3.
G.f.: 1/(1x3*x^3).
a(n) = sum{k=0..floor(n/2), C(n2k, k)3^k}  Paul Barry, Nov 18 2003
G.f.: W(0)/2, where W(k) = 1 + 1/(1  x*(1 + 3*x^2)/(x*(1 + 3*x^2) + 1/W(k+1) )); (continued fraction).  Sergei N. Gladkovskii, Aug 28 2013
Starting (1 + x + 4x^2 + ...), is the INVERT transform of (1 + 3x^2).  Gary W. Adamson, Mar 27 2017


MAPLE

seq(add(binomial(n2*k, k)*3^k, k=0..floor(n/3)), n=0..34);  Zerinvary Lajos, Apr 03 2007


MATHEMATICA

a[0]=a[1]=a[2]=1; a[n_] := a[n]=a[n1]+3a[n3]; Table[a[n], {n, 0, 34}]
LinearRecurrence[{1, 0, 3}, {1, 1, 1}, 37] (* Robert G. Wilson v, Jul 12 2014 *)


PROG

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 3, 0, 1]^n*[1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
(MAGMA) I:=[1, 1, 1]; [n le 3 select I[n] else Self(n1)+3*Self(n3): n in [1..40]]; // Vincenzo Librandi, Mar 28 2017


CROSSREFS

Partial sums of A052900. Also A052900/3.
Cf. A000930, A006130, A001045.
Sequence in context: A227686 A161863 A102649 * A275176 A024726 A024948
Adjacent sequences: A084383 A084384 A084385 * A084387 A084388 A084389


KEYWORD

easy,nonn


AUTHOR

Merrill Jensen (mpjensen(AT)mninter.net), Jun 23 2003


EXTENSIONS

Edited by Dean Hickerson, Jun 24 2003
Recurrence appended to the name by Antti Karttunen, Mar 28 2017


STATUS

approved



